Theorem dfdm3 4854

Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfdm3	|- dom A = { x | E. y <. x , y >. e. A }

Distinct variable group:   x, y, A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 4674	. 2 |- dom A = { x | E. y x A y }
2		df-br 4035	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1619	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2312	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2217	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1364   E.wex 1506   e. wcel 2167   {cab 2182   <.cop 3626   class class class wbr 4034   dom cdm 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-br 4035  df-dm 4674

This theorem is referenced by:  csbdmg  4861